Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.3% → 96.8%

Time: 2.5s

Alternatives: 3

Speedup: 1.2×

• 43.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))

C

double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

Python

def code(x, y):
	return ((x * x) + ((x * 2.0) * y)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) + ((x * 2.0) * y)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))

C

double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

Python

def code(x, y):
	return ((x * x) + ((x * 2.0) * y)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) + ((x * 2.0) * y)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + 2 \cdot y\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x (+ x (* 2.0 y))) (* y y)))

C

double code(double x, double y) {
	return (x * (x + (2.0 * y))) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * (x + (2.0d0 * y))) + (y * y)
end function

Java

public static double code(double x, double y) {
	return (x * (x + (2.0 * y))) + (y * y);
}

Python

def code(x, y):
	return (x * (x + (2.0 * y))) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(x + Float64(2.0 * y))) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * (x + (2.0 * y))) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

   1. +-commutative 96.5%

      \[\leadsto \color{blue}{\left(\left(x \cdot 2\right) \cdot y + x \cdot x\right)} + y \cdot y \]

   2. associate-*l* 96.5%

      \[\leadsto \left(\color{blue}{x \cdot \left(2 \cdot y\right)} + x \cdot x\right) + y \cdot y \]

   3. distribute-lft-out 99.2%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + x\right)} + y \cdot y \]

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + x\right)} + y \cdot y \]

4. Final simplification 99.2%

   \[\leadsto x \cdot \left(x + 2 \cdot y\right) + y \cdot y \]

Alternative 2: 54.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ y \cdot y + y \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (* y (* x 2.0))))

C

double code(double x, double y) {
	return (y * y) + (y * (x * 2.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * y) + (y * (x * 2.0d0))
end function

Java

public static double code(double x, double y) {
	return (y * y) + (y * (x * 2.0));
}

Python

def code(x, y):
	return (y * y) + (y * (x * 2.0))

Julia

function code(x, y)
	return Float64(Float64(y * y) + Float64(y * Float64(x * 2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) + (y * (x * 2.0));
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

2. Taylor expanded in x around 0 60.4%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} + y \cdot y \]
3. Step-by-step derivation

   1. associate-*r* 60.4%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot y} + y \cdot y \]

   2. *-commutative 60.4%

      \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot y + y \cdot y \]

   3. *-commutative 60.4%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot 2\right)} + y \cdot y \]

   4. *-commutative 60.4%

      \[\leadsto y \cdot \color{blue}{\left(2 \cdot x\right)} + y \cdot y \]

4. Simplified 60.4%

   \[\leadsto \color{blue}{y \cdot \left(2 \cdot x\right)} + y \cdot y \]

5. Final simplification 60.4%

   \[\leadsto y \cdot y + y \cdot \left(x \cdot 2\right) \]

Alternative 3: 15.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (* x 2.0)))

C

double code(double x, double y) {
	return y * (x * 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * (x * 2.0d0)
end function

Java

public static double code(double x, double y) {
	return y * (x * 2.0);
}

Python

def code(x, y):
	return y * (x * 2.0)

Julia

function code(x, y)
	return Float64(y * Float64(x * 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = y * (x * 2.0);
end

Wolfram

code[x_, y_] := N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

2. Taylor expanded in x around 0 60.4%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} + y \cdot y \]
3. Step-by-step derivation

   1. associate-*r* 60.4%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot y} + y \cdot y \]

   2. *-commutative 60.4%

      \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot y + y \cdot y \]

   3. *-commutative 60.4%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot 2\right)} + y \cdot y \]

   4. *-commutative 60.4%

      \[\leadsto y \cdot \color{blue}{\left(2 \cdot x\right)} + y \cdot y \]

4. Simplified 60.4%

   \[\leadsto \color{blue}{y \cdot \left(2 \cdot x\right)} + y \cdot y \]

5. Taylor expanded in y around 0 16.7%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
6. Step-by-step derivation

   1. associate-*r* 16.7%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot y} \]

   2. *-commutative 16.7%

      \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot y \]

   3. *-commutative 16.7%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot 2\right)} \]

   4. *-commutative 16.7%

      \[\leadsto y \cdot \color{blue}{\left(2 \cdot x\right)} \]

7. Simplified 16.7%

   \[\leadsto \color{blue}{y \cdot \left(2 \cdot x\right)} \]

8. Final simplification 16.7%

   \[\leadsto y \cdot \left(x \cdot 2\right) \]

Developer target: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (+ (* y y) (* (* x y) 2.0))))

C

double code(double x, double y) {
	return (x * x) + ((y * y) + ((x * y) * 2.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + ((y * y) + ((x * y) * 2.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * x) + ((y * y) + ((x * y) * 2.0));
}

Python

def code(x, y):
	return (x * x) + ((y * y) + ((x * y) * 2.0))

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(Float64(x * y) * 2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + ((y * y) + ((x * y) * 2.0));
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023312 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))